Daniel Cloutier and Joshua Holden

Abstract

The discrete logarithm is a problem that surfaces frequently in
the field of cryptography as a result of using the transformation
x↦gx modn. Analysis
of the security of many cryptographic algorithms depends on the assumption that it is
statistically impossible to distinguish the use of this map from the use of a randomly
chosen map with similar characteristics. This paper focuses on a prime modulus,
p,
for which it is shown that the basic structure of the functional graph
produced by this map is largely dependent on an interaction between
g and
p−1. We
deal with two of the possible structures, permutations and binary functional graphs.
Estimates exist for the shape of a random permutation, but similar estimates must
be created for the binary functional graphs. Experimental data suggest that both the
permutations and binary functional graphs correspond well to the theoretical
predictions.